The necessary and sufficient condition for maximal estimates to hold is that \(q\) –sums of eigenvalues of the Levi eigenvalues dominate the trace of the Levi form ([1]}, Théorème 3.1 for \(q=1\) , [2]}, Théorème 3.7 for \(q>1\) ). More precisely, this condition means that there is a constant \(C\)  such that for any point \(p\in b\Omega \)  and \(j_{1},\cdots ,j_{q}\)  with \(1\le j_{1}<\cdots <j_{q}\le (n-1)\) , it holds that \(C(\lambda _{1}(p)+\lambda _{2}(p)+\cdots +\lambda _{n-1}(p)) \le \lambda _{j_{1}}(p)+\cdots +\lambda _{j_{q}}(p)\) , where \(\lambda _{1}(p)\) , \(\cdots \) , \(\lambda _{(n-1}(p)\)  denote the eigenvalues of the Levi from at \(p\) , listed with multiplicity These eigenvalues are independent of the basis as long as they are taken with respect to an orthonormal basisIn [3]}, footnote 1, the authors point out that thanks to a theorem of Ostrowski ([4]}, Theorem 4.5.9) the comparability condition remains independent of the basis even when the orthonormality requirement is dropped.. Because \(\Omega \)  is pseudoconvex, all Levi eigenvalues are non negative, and the previous condition is easily seen to be equivalent to the following one: there exists a constant \(C\)  such that for any pair of \(q\) –tuples \((j_{1}, \cdots , j_{q})\)  and \((k_{1}, \cdots , k_{q})\) , it holds that \(\lambda _{j_{1}}(p)+\cdots +\lambda _{j_{q}}(p) \le C(\lambda _{k_{1}}(p)+ \cdots +\lambda _{k_{q}}(p))\)  (by an argument similar to the one in the proof of Lemma REF  below). We say for short that \(q\) –sums of the Levi eigenvalues of \(\Omega \)  are comparable. Note that for \(q=(n-1)\) , there is only one \(q\) –sum, and the comparability condition trivially holds. In particular, the condition trivially holds for domains in \(\mathbb {C}^{2}\)  and \(q=1\) .
